Skip to main content
SpringerLink
Log in

Banach geometry of arbitrage free markets

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

The article presents a description of geometry of Banach structures imitating arbitrage absence type phenomena in the models of financial markets. In this connection we uncover the role of reflexive subspaces (replacing classically considered finite-dimensional subspaces) and plasterable cones. A number of new geometric criteria for arbitrage freeness is proven; and an alternative description of a martingale measure existence not exploiting dual objects is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Bakhtin, I.A.: Cones in Banach spaces, part 1. — Voronezh. ped. inst., (1975) (in Russian)

  3. Bourbaki, N.: Espaces Vectoriels Topologiques (Elements de Mathematique). Paris, Hermann & C, Editeurs (1956)

    Google Scholar 

  4. Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer Finance, Berlin (2006)

    MATH  Google Scholar 

  5. Dunford, N., Schwartz, J.: Linear Operators Part 1: General Theory. Interscience Publishers, New York, London (1958)

    MATH  Google Scholar 

  6. Follmer, H., Schied, A.: Stochstic Finance: An Introduction in Discrete Time. Walter de Guyter, Berlin, New York (2004)

    Book  Google Scholar 

  7. Krasnosel’skij, M.A.: Positive solutions of operator equations. P. Noordhoff, Groningen (1962)

    Google Scholar 

  8. Krasnosel’skij, M.A., Lifshits Je, A., Sobolev, A.V.: Positive Linear Systems: The Method of Positive Operators. Heldermann, Berlin (1989)

    Google Scholar 

  9. Pascucci, A.: PDE and Martingale Methods in Option Pricing. Springer, Berlin (2011)

    Book  Google Scholar 

  10. Schachermayer, W.: The fundamental theorem of asset pricing. Encyclopedia of Quantative Finance 2, 792–801 (2010)

    Google Scholar 

  11. Zabreiko, P.P., Lebedev, A.V.: Banach geometry of financial markets models. Doklady Mathematics, V 95, 164–167 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Bialystok, Białystok, Poland

    A. V. Lebedev

  2. Belarusian State University, Minsk, Belarus

    A. V. Lebedev & P. P. Zabreiko

Authors
  1. A. V. Lebedev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. P. Zabreiko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Lebedev.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lebedev, A.V., Zabreiko, P.P. Banach geometry of arbitrage free markets. Positivity 25, 679–699 (2021). https://doi.org/10.1007/s11117-020-00782-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-020-00782-6

Keywords

Mathematics Subject Classification

Search

Navigation